Limit of a function | IIT JAM 2017

Try this problem from IIT JAM 2017 exam. It deals with evaluating Limit of a function.

Limit of a Function | IIT JAM 2017 | Problem 8

Let

$f(x)=\frac{x+|x|(1+x)}{x} \sin \left(\frac{1}{x}\right), \quad x \neq 0$

Write $L=\displaystyle\lim_{x \to 0^{-}} f(x)$

and $R=\displaystyle\lim_{x \to 0^{+}} f(x) .$

Then which one of the following is true?

$L$ exists but $R$ does not exist
$L$ does not exist but $R$ exists
Both $L$ and $R$ exist
Neither $L$ nor $R$ exists

Key Concepts

Real Analysis

Function

Limit

Check the Answer

Answer: $L$ exists but $R$ does not exists

IIT JAM 2017 , Problem 8

Try with Hints

Given that, $f(x)=\frac{x+|x|(1+x)}{x} \sin \left(\frac{1}{x}\right), \quad x \neq 0$

therefore,

$f(x)=1+\frac{|x|}{x}(1+x) \sin \left(\frac{1}{x}\right), \quad x \neq 0$

$f(x)=\bigg\{\begin{array}{cc} (2+x) \sin \left(\frac{1}{x}\right), & , x>0 \\ -x \sin \left(\frac{1}{x}\right), & x<0 \\ \end{array}$

Let, $L=\displaystyle\lim_{x \to 0^{-}} f(x)$

and , $R= \displaystyle \lim_{x \rightarrow 0^{+}} f(x) .$

Now,

$L= \displaystyle\lim_{x \to 0^{-}} f(x)$

$\quad = \displaystyle\lim_{x \to 0^{-}} -x \sin \left(\frac{1}{x}\right)$

$\quad = -\displaystyle\lim_{x \to 0^{-}} x \sin \left(\frac{1}{x}\right)$

Theorem : If $D \subset \mathbb R$ and $f,g : D \to \mathbb R$ . Let $c \in D$ . If f is bounded on $N'(c)\cup D$ and $\displaystyle\lim_{x \to c} g(x)=0$ , then $\displaystyle\lim_{x \to c}(f.g)(x)=0$ .

Now , $\sin \left(\frac{1}{x}\right)$ is bounded in $\mathbb R - \{0\}$ and $\displaystyle\lim_{x \to 0^{-}} x=0$ , then $\displaystyle\lim_{x \to 0^{-}} f(x)$ exists and equal to $0$ .

But,

$R=\displaystyle\lim_{x\to 0^{+}}f(x)$

$\quad = \displaystyle\lim_{x\to 0^{+}} (2+x) \sin \left(\frac{1}{x}\right),$

$\quad= \displaystyle\lim_{x\to 0^{+}} 2\sin \left(\frac{1}{x}\right) + x \sin \left(\frac{1}{x}\right)$

$\lim_{x \to 0^{+}} \sin \left(\frac{1}{x}\right)$ does not exists [Why?]

Then $L$ exists but $R$ does not.